NUMERICAL CALCULATION METHOD FOR MAGNETIC FIELDS IN THE VICINITY OF CURRENT-CARRYING CONDUCTORS

Numerical calculation method for magnetic fields in the vicinity of current-carrying conductors

Gustav Gärskog

Teknisk- naturvetenskaplig fakultet UTH-enheten

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Numerical calculation method for magnetic fields in the vicinity of current-carrying conductors

Gustav Gärskog

This thesis aims to develop a calculation method to determine the magnetic field magnitudes in the vicinity of power lines, i.e. both buried cables and overhead lines. This is done through the numerical use of Biot Savart's law where the conductors are approximated by a series of straight segment elements that each contribute to the overall field strength at the field point. The method is compared to two real cases and to the exact integral solution. Also, a review of some of the research material regarding electromagnetic fields from power lines and claims of adverse health effects due to these fields is conducted.

Results show that the numerical error is dependent on the segmentation degree of the conductors and the mathematical model is inaccurate close to the conductor. The calculations show slightly higher field

magnitudes than the previous survey done by WSP (Williams Sale Partnership) far away from the source and slightly lower at the center conductor. This may be due to the excluded induction in the shield wires and differences in actual conductor coordinates.

ISSN: 1654-7616, UPTEC E** ***

Examinator: Mikael Bergkvist Ämnesgranskare: Marcus Berg Handledare: Ivan Barck

Sammanfattning

Magnetiska fälts påverkan på människans hälsa har länge debatterats. Även en stor mängd forskning har genomförts på området. I synnerhet har ett samband mellan barn- leukemi och lågfrekventa magnetiska fält visat sig vara troligt och varit flitigt debatterat.

Även andra former av cancer och neurologiska sjukdomar har visat vissa samband med magnetiska fält. Inga tydliga bevis har dock fastställts.

Då inget lagstadgat gränsvärde idag finns tillämpas försiktighetsprinsipen, vilket in- nebär att ägaren av nätkoncetion ska förebygga eventuell risk för människors hälsa inom rimliga ekonomiska ramar. Praxis har, i väntan på mer forskningsunderlag, under senare år varit ett riktvärde på 0, 4 µT för platser där människor bor och vistas varaktigt. Delar av forskningsmatrealet rörande magnetiska fälts eventuella hälsoeffekter har studerats i denna avhandling i syfte att ge en bakgrund och bedöma relevansen av det riktvärde som idag används för kraftledningar och kraftkablar.

Denna avhandlings huvudsyfte är att utveckla ett beräkningsprogram för att bestämma magnetfältets storheter i närmiljön till kraftledningar, d.v.s. både kablar och luftled- ningar. Detta görs med hjälp av Biot Savarts lag där ledarna approximeras av en serie raka segmentelement som var för sig bidrar till den totala fältstyrkan i fältpunkten. Sum- man av fältbidragen blir således den totala fältstyrkan i en given fältpunkt. Eftersom det är långtidsexponering som är av vikt vid dessa beräkning används årsmedelströmmar.

Metoden jämförs med två reella fall och en något förenklad integrallösning för att ge en uppfattning om metodens exakthet. Vissa delar av problemet har gjorts avkall på. Induk- tionen i neutralledarna (topplinorna) tas inte med i beräkningsmodellen i detta skede, vilket den har gjorts i referens materialet. Avvikelser har observerats i de beräknade magnetfältsmagnituderna framförallt nära mittfasen hos de luftledningssystem som har jämförts, vilket kan vara en följd av induktionsbidraget. Svårigheter att återskapa de exakta beräkningsförutsättningar som använts vid WSP:s beräkningar kan också bidra till eventuella skillnader i resultaten. Resultaten visar att det numeriska felet är beroende av ledningarnas segmenteringsgrad och att den matematiska modellen är inexakt nära ledaren. Beräkningarna har en något högre fältmagnitud än den tidigare undersökningen som gjorts av WSP långt ifrån källan och något lägre vid mittfasen nära källan. Detta kan bero på avsaknaden av induktionsbidraget från topplinorna och skillnader i ledarkoordinater. Abstract

Dissertation in partial fulfillment of the requirements for the degree of:

Master of Science in Engineering, 300 credits Specialization; Electrical Engineering

Uppsala University Department of Electricity Science

[Gustav Gärskog]

[07 05 2018]

Master of Science in Engineering, 300 credits Specialization; Electrical Engineering

Uppsala University Department of Electricity Science

Approved by

Supervisor, Ivan Barck, WSP Subject reviewer, Marcus Berg

Examiner, Mikael Bergkvist

[07-05-2018]

Acknowledgements

I would like to give a sincere thank you to Ivan Barck who has helped me tremendously through this process. Also a big thanks to the team at Williams Sale Partnership (WSP) elkraft that has provided a home away from home during this experience. I would also like to direct a big thank you to Marcus Berg that has helped me greatly to elevate this thesis to its current state.

Table of Contents

Acknowledgements I

List of Tables IV

List of Figures V

List of Acronyms VII

1 Introduction 1

1.1 Background . . . 1

1.2 Health effects . . . 3

1.2.1 Physical interaction between biological tissue and fields from power lines . . . 4

1.2.2 Physical effects of electric fields in the body . . . 4

1.2.3 Basic interaction of magnetic fields in the body . . . 4

1.2.4 Neurodegenerative disorders . . . 5

1.2.5 Cardiovascular disorders . . . 5

1.2.6 Immune system and hematology . . . 6

1.2.7 Reproduction and development . . . 6

1.2.8 Cancer . . . 7

1.3 Shielding of magnetic fields . . . 7

1.4 Cases . . . 8

1.4.1 Integral solution . . . 8

1.4.2 Bladsjön . . . 8

1.4.3 Grundfors . . . 10

1.5 Underground cables . . . 11

2 Methodology 12 2.1 Biot Savart’s law . . . 12

2.2 Simplifications . . . 13

2.3 Calculation method . . . 14

2.3.1 Yearly current average . . . 14

2.3.2 Numerical calculation method . . . 14

3 Results 15 3.1 Special case evaluation : Straight conductor . . . 15

3.2 Special case evaluation: Hyperbolic conductor . . . 19

3.3 Bladsjön . . . 21 3.4 Grundfors . . . 26

4 Discussion and analysis 31

4.1 Medical effects . . . 31 4.2 Calculation method . . . 31

5 Conclusion 33

Literature 35

A Appendix 37

A.1 [Magnetic B-field from three phase hyperbolic conductors] . . . 38 A.2 Bladsjön . . . 39 A.3 Grundfors . . . 40

List of Tables

Table 1.1: Magnetic fields produced by some everyday household appliances at different distances from the source [3] . . . 2 Table 3.1: Comparison of the exact integral and the numeric solution method for

different segment lengths at field point (50,0,0). . . 16 Table 3.2: Comparison of the exact integral and the approximate solution for

different segment lengths at field point (20,0,0). . . 17 Table 3.3: Comparison between the accuracy at the closest, optimal and 100 m

horizontal distance from the conductor . . . 20 Table 3.4: Numerical field calculation for houses with conductor segment lengths

at 5 m and 1 m. . . 23

List of Figures

Figure 1.1: parallel single-circuit power lines, note the hyperbolic line sag be- tween towers. (Magnus Manske|Cc-by-3.0.) . . . 1 Figure 1.2: The magnetic field strength at 1.5 m over ground from a three-phase

circuit with three different currents. The distance between the phases is 10 m. The currents are chosen to be representative of the power levels in the rest of this thesis. . . 3 Figure 1.3: Suggested changes for the site Bladsjön outside of Åsbro and its

existing 400 kV power line. Residential homes are marked with green outlining. The yellow lines represents the existing power lines, (NW- SE, 400 kV, 450 A), (N-S, 130 kV, -200 A). Alternative 3B is light blue, 3C is dark blue, 3D is green and 4A is purple. . . 9 Figure 1.4: Overhead image of the site at Grundfors. The two green circles

indicate houses of interest, magenta lines represent the altered section of the power lines and the red outline the new location of the switch yard. The east green line is the cross section compared using the reference material and the calculation method. . . 10 Figure 3.1: Plot showing how the error of the numerical result depends on wire

segmentation and how odd and even numbers of segments affect the relative error. . . 17 Figure 3.2: 24 segments: the ratio between calculated numerical result and exact

solution vs distance from source. The total length of the conductor is 500 m. . . 18 Figure 3.3: 25 segments: the ratio between calculated numerical result and exact

solution vs distance from source. The total length of the conductor is 500 m. . . 18 Figure 3.4: 100 segments: the ratio between calculated numerical result and exact

solution vs distance from source. The total length of the conductor is 500 m. . . 19 Figure 3.5: 101 segments: the ratio between calculated numerical result and exact

solution vs distance from source. The total length of the conductor is 500 m. . . 19 Figure 3.6: The ratio between 5 m segment length and the reference 0.1 m . . . . 20 Figure 3.7: The ratio between 20 m segment length and the reference 0.1 m . . . 20

Figure 3.9: 2D representation of the site Bladsjön with relevant houses. The red line corresponds to the cross section in figures 3.10 and 3.11. The black rectangle represents the calculation area used in figure 3.12 . . 22 Figure 3.10: Graph of the magnetic field near house 1:71 in WSP’s report for

Bladsjön. The section is indicated in Figure 3.8. The field calculation height is 1.5 m above ground level at 1:71. The origin indicates the center phase. . . 23 Figure 3.11: Calculation of the B-field for the same section as in Figure 3.10, using

the algorithm developed by the author. . . 24 Figure 3.12: Graph of result for site Bladsjön for house 1:71 altitude above sea level 24 Figure 3.13: Contour graph of three B-field magnitudes for site Bladsjön at house

1:71 height above sea level. The house 1:71 is indicated by the blue circle. Contour levels are given in µT. . . . 25 Figure 3.14: 2D representation of the site Grundfors with relevant houses. The

yellow line represents the cross section in figure 3.15 and 3.16. The black rectangle represents the area used in figure 3.18. . . 26 Figure 3.15: B-field magnitude as a function of the horizontal distance from the

center phase of line UL1 S1-3. The curve corresponds to the cross section towards the Eastern house in figure 3.14. Reference curve from WSP. . . 27 Figure 3.16: Cross section field strength plot for the same cross section as in figure

3.10, calculated result . . . 28 Figure 3.17: Contour plot for the Grundfors site. The east house is too far outside

the relevant area and hence not included. The western house is shown as a blue circle. Contour levels are given in µT. . . . 29 Figure 3.18: Calculated B-field magnitude for the Grundfors site. Peaks corre-

spond to points, where the conductors are closest to ground. The field is calculated at 1.5 m above the ground plane. Compare this plot with figure 3.14 . . . 30

List of Acronyms

ELF Extremely Low Frequency(0 Hz−100 kHz)

SVK Svenska Kraftnät

WSP Williams Sale Partnership ALS Amyotrophic Lateral Sclerosis

MS Multiple Sclerosis

AD Alzheimer’s Disease

WHO World Health Organization

CV Cardiovascular

HRV Heart Rate Variability AMI Acute Myocardial Infarction

EI Energimarknadsinspektionen (Swedish energy market inspectorate)

FEM Finite Element Method

1 Introduction

1.1 Background

In modern society, energy demand keeps increasing and shows no sign in declining [1].

To supply this demand power systems are essential to transport the converted energy from the sources, which often are located at remote locations, far away from the main consumers. As a direct consequence, magnetic and electric fields are induced around the power lines. The magnetic field strength is dependent on the currents flowing through the conductors at any given time. As the Swedish power grid operates at 50 Hz, the resulting electromagnetic field has the same frequency.

Figure 1.1: parallel single-circuit power lines, note the hyperbolic line sag between towers. (Magnus Manske|Cc-by-3.0.)

These magnetic end electric fields are everywhere in our work and living environments.

Every conductor that has a current flowing through it will produce a electromagnetic

field around it. Therefore, it is important to understand the effects these fields have on biological organisms. There have been claims that these fields may have some adverse health effects. In light of these claims, safety precautions in the form of a guideline limit at 0.4 µT, for residential homes, have been put in place pending conclusive proofs.

This value is a recommended limit by Energimarknadsinspektionen (Swedish energy market inspectorate) (EI) for new and rebuilt power lines. If the magnetic field strength exceeds 0.4 µT then preemptive measures to lower the field strength will have to be pre- sented to EI. EI will then decide if the cost to lower magnetic field exposure is reasonable.

In this thesis, the consequences and health effects of these fields are discussed and a calculation method developed for predicting magnetic field strength near power lines at an early stage of project planning. The magnetic fields that are the basis of this thesis are categorized by World Health Organization (WHO) as Extremely Low Frequency (_{0 Hz}−_{100 kHz})(ELF) [2]. These fields are ubiquitous in our work and living envi- ronment and cannot be completely eradicated. For the scope of this thesis, only the power grid distribution system is considered in the calculations. In order to illustrate the relative strength of magnetic fields from power lines, field strengths of some everyday household appliances are listed in table 1.1

Table 1.1: Magnetic fields produced by some everyday household appliances at different distances from the source [3]

Appliance 0.1 m 0.5 m 1 m

Television 1.5−4 µT 0.2−1 µT 0.1−0.2 µT Stove 1−3 µT 0.1−0.6 µT 0.05−0.2 µT Hairdryer 0.5−_{12 µT 0.1}−_{0.3 µT 0.05}−_{0.1 µT} Vacuum cleaner 15−35 µT 0.4−1.5 µT 0.1−0.5 µT

In table 1.1, many of the appliances exceed the limit for fields from power lines. Im- portant to remember, however, is that most of these appliances mainly are used short periods of time and not continuously, as in the case of power lines, based on yearly aver- age current magnitudes. The instantaneous magnitudes generated by power lines can still be higher or lower at any given time. This, nevertheless, shows that the occurrence of magnetic fields is not an isolated event in the context of power transmission lines and that the strengths of these fields considered in this thesis are not much stronger than some common items that are considered safe to use. The field strength also declines rapidly as the distance gets larger from the source. Figure 1.2 shows magnetic fields from a three-phase circuit as a function of the distance from the center phase conductor.

Figure 1.2: The magnetic field strength at 1.5 m over ground from a three-phase circuit with three different currents. The distance between the phases is 10 m. The currents are chosen to be representative of the power levels in the rest of this thesis.

The main goal of this thesis is to produce a working tool for calculation of these magnetic fields in the vicinity of both underground cables and overhead transmission lines using MATLAB. This is done to easily determine where the limit is located in relation to the power line. The most important point of interest is the nearest point of residential homes at the height of 1.5 m, above the ground plane, as this can be considered as the average height of a human torso, which is the part of the body that is subjected to the highest induced currents.

1.2 Health effects

For many people, the fear and belief that magnetic fields are harmful is very real, even though no conclusive proof of any adverse effect has been produced. There are, however, many claims about adverse health effects, and although inconclusive, research suggests that some harmful effects may be caused by these fields, although to a limited extent. As a result it has been necessary to take preemptive measures when planning for the new grid and rebuilding the old one, in the form of a guide value. Some of the health effects that have been researched and may be triggered by magnetic fields are [2]:

1. Neurodegenerative disorders 2. Cardiovascular disorders

3. Immune system and heamatology 4. Reproduction and development 5. Cancer

The most controversial effect and the one showing most conclusive evidence is the correlation between ELF magnetic fields and childhood leukemia.

1.2.1 Physical interaction between biological tissue and fields from power lines

Power transmission lines produce both an ELF electrical field and an ELF magnetic field.

"ELF" stands for "Extreme Low Frequency", and is used specifically by WHO to denote the frequency interval of 0-100 kHz. These fields produce physical effects in organisms.

Although the interactions are different, both fields have the ability to give rise to currents in the body tissue. The strength of the induced electric fields and, in turn, currents is to some extent dependent on the body mass, area and the curvature of the different body parts [2][3].

1.2.2 Physical effects of electric fields in the body

A body placed in an electric field will significantly perturb the electric field depending on the body size and shape and the way the body is connected to the ground. With a perfect connection to the ground through the feet a larger current will flow through the body. The currents are generated through oscillating charges on the skin that induce currents inside the body [2].

1.2.3 Basic interaction of magnetic fields in the body

External magnetic fields that a biological organism comes in contact with are not re- stricted by the tissue. The amount of magnetic materials present is too small to give a noteworthy contribution. Therefore, the relative permeability can be considered equal to that of free space [2]. As an ELF magnetic field passes through the body with varying strength, it will induce a varying electric potential distribution (inside and on the body), described by Faraday’s induction law:

I

C

~_E·d~_s= −^dΦ

dt (1.1)

where C is an arbitrary closed path in space. Φ is the magnetic flux through the closed path C, and is described by the equation:

Φ=

"

S

~B(t) ·d_A~ _(1.2)

1.2.4 Neurodegenerative disorders

The main suggested links between magnetic fields and neurodegenerative disorders are Alzheimer’s disease, Parkinson’s disease, Amyotrophic Lateral Sclerosis (ALS) and Multiple Sclerosis (MS) [2][5]. Many studies have been conducted on the subject, with the main focus on ALS as it shows the most likely correlation with an electrically intense work environment.

Amyotrophic Lateral Sclerosis

ALS is characterized by motor dysfunction muscle atrophy. The causes of the disease have not been completely established but can either be environmental factors or in some cases be genetically inherited. The most interesting environmental factor for this report is cases of electrical trauma. WHO suggests an increased risk of ALS for workers in electrical occupations [2]. This increase could be a result of the heightened risk of shock in these work environments rather than the higher electromagnetic field strength exposure [2][5]. Links with ELF magnetic field exposure are not explored in detail.

However, some of the subjects that developed the disease had previous experience of strong electrical shock. Since magnetic fields also produce a current in the human body, although much smaller in amplitude, the hypothesis cannot be completely disproved.

Parkinson’s Disease and Multiple Sclerosis

Suggested from relevant studies reviewed in [2], these two diseases show the least likely link to ELF fields, and may be considered unlikely as a consequence of ELF magnetic field exposure.

Alzheimer’s Disease

Very limited evidence of correlation between ELF magnetic field exposure and Alzheimer’s Disease (AD) has been observed. The focus of the overall material is occupational envi- ronments. Some studies present a link but the conclusion is not completely confirmed in the rest of the material covered by the WHO report [2]. The conclusion of the WHO study is that the evidence is inadequate as proof.

1.2.5 Cardiovascular disorders

There are concerns that ELF magnetic field exposure may have chronic effects on the Cardiovascular (CV) system. Changes have been observed in switchyard operators

during 1960 through 1970. As in the case of ALS, current flow through the body due to electric shock seems necessary for any major CV effects. It is not known if these effects also can occur through the exposure to very strong ELF magnetic fields. The conclusion drawn is that this is unlikely to appear at commonly encountered levels [2]. Later studies focus more on the acute CV events due to Heart Rate Variability (HRV). One of these studies is "Occupational Exposure to Extremely Low Frequency Magnetic Fields and Mortality" by Håkansson [6]. Four different CV conditions were considered

1. Arrhythmia-related death

2. Acute Myocardial Infarction (AMI) 3. Atherosclerosis

4. Ischemic heart disease other than AMI

The only condition that showed a slight increase was Arrythmia-related death. AMI and atherosclerosis showed no direct effect. Ischemic heart disease, other than AMI, showed decreased risk. The exposure levels used in the study were low, medium high and very high corresponding to field levels of<0.1µT, 0.10–0.19µT, 0.20–0.29µT and

≥0.3µT respectively. The associations between ELF magnetic fields and these diseases are therefore determined to be weak.

1.2.6 Immune system and hematology

Not much research has been conducted on the subject of immune system and hema- tological effects due to ELF magnetic field exposure. Some changes in the numbers of natural killer cells and white blood cells, that are critical to the immune system, have been observed, but to a limited extent. In some cases the natural killer cell numbers were even higher after exposure, which suggests that ELF magnetic fields have no significant consequence on the immune system.

Even fewer studies have been carried out on hematological changes, and no evidence of adverse effects have been found [2]. Results in [7] propose that cells exposed to a 60 Hz electric field inhibited the T-lymphocytes that attack cancer cells.

1.2.7 Reproduction and development

Radio frequency magnetic fields effect on reproduction is a widely spread hypothesis and have shown evidence of reducing semen count, mobility, viability and normal morphology. Most research focus is on the link between cell phone use and these effects [8],[9]. Also extended laptop use and wifi in close vicinity to the subject’s testes are thought to lead to similar effects. The question if magnetic fields from power lines pose a risk is therefore relevant. Also the female reproduction is explored and according to WHO some effects have been observed in the form of a higher miscarriage risk [2].

1.2.8 Cancer

Most of the overall research focus has been put on proving a link between ELF magnetic fields and possible increase in cancer. The link between exposure to these fields and childhood leukemia has been found in a number of studies. The ELF magnetic fields are unlikely, due to their low energy, to directly cause genetic mutations but may contribute to this development indirectly. The minimum field strength needed for direct damage to DNA has been believed to be at 50 µT, although this value may be revised due to findings in more recent reports. The general belief is nevertheless that the magnetic fields from transmission lines are not the direct cause of the mutations. Some theo- ries have been presented as to why a higher risk has been observed. One example of this is that the magnetic fields may cause potential differences in conducting materials that when touched would conduct current through the body with sufficient energy to affect the bone marrow. The currents are also less perturbed in the body of a child.

Another hypothesis is similar to that presented for AD, as melatonin production may be effected by the exposure. However, no proof of these claims have been found so far [2].

In [10] no increase was registered for children living over the threshold 0.4 µT, but some increase was noted for the children living at levels≥0.4 µT. The use of 0.4 µT as a limit might very well be a consequence of this study’s result. The relative risk assessment for children living over 0.4 µT was concluded to be twice that of children living below the exposure guide value and unlikely to be due to random variation.

For adults, breast cancer as well as brain cancer and leukemia have been considered plausible effects [2]. Melatonin production changes are thought to be the catalyst in the case of breast cancer. WHO conclude that the overall studies are negative and that the evidence of an association between ELF magnetic fields and breast cancer is weak. The findings for brain cancer are considered inadequate.

1.3 Shielding of magnetic fields

There are methods with which the fields surrounding the transmission lines can be reduced. More and more cables are buried, which makes it possible to keep the conduc- tors closer together or even to have all three phase conductors in one cable. Since the instantaneous current in a balanced three phase circuit is zero for any given time, the magnetic field from such a cable will be very small, as all three phases approximately coincide in space. When considering separate phase conductors, the magnetic field will be significantly smaller when the conductors are closer together, resulting in more of the

vector contributions from each phase canceling each other.

Another method of limiting the magnetic fields is to place the circuits in a conduct- ing or ferromagnetic shell, for example made from aluminum or iron. Results presented in [11] show substantial dampening. It is concluded that aluminum is the better choice of the two. The main problem concerning both methods is the substantial cost of materials.

This alone makes the method only viable in very specific cases. The fact that substantial losses will appear in the material as well, producing heat, needs to be taken into account when dimensioning such a system.

Both burying cables and shielding the magnetic fields will lead to a large cost. A simpler way to reduce fields from the circuits is optimal phase ordering of the conduc- tors in space. This method can be utilized when the overall system contains more than one three-phase circuit [11], [12].

1.4 Cases

1.4.1 Integral solution

The solution method will be compared with the exact solution of Biot Savart’s law, with the current approximated as a line current. This is done in order to assess the strengths and weaknesses of the method and also to determine the error of the numerical result for the case of a straight conductor. The error will depend on the segmentation of the conductors. For smaller segment lengths the approximate solution converges on the exact integral solution, but the calculation complexity decreases greatly with fewer and longer elements. This makes it necessary to choose an optimal segment length for the given situation and the desired accuracy.

The exact integral becomes difficult to solve analytically for more complex cases. There- fore, the simple case of a single straight conductor is investigated first. Comparing the exact solution with the numerical solution gives some insight into the strengths and weaknesses of the method. This example serves as a useful reference mostly for cables underground as these can be more accurately approximated as straight conductors.

For the cases with overhead power lines, that are described in the following sections, estimating the error is a more complex task. Still, an effort was made by using a finely segmented conductor calculation as a reference.

1.4.2 Bladsjön

For assessment of the finished method, results for two real cases were produced and compared to calculations made by WSP. The first case is the site of Bladsjön outside

site have been determined in a previous survey and will serve as a reference when doing the same calculations with the produced MATLAB script. Global coordinates for the three phase systems are calculated from the tower positions as hyperbolic curves with data points every meter horizontally along the conductor. The actual segment length is therefore a little bit longer than 1 m and not completely uniform along the conductor.

For the accuracy of the calculation the distance between data points needs to be kept at a minimum. The time of calculation is also highly dependent on the number of data points which increases the calculation time for the complete system at higher resolution.

Suggested changes for the site can be viewed in figure 1.3

Figure 1.3: Suggested changes for the site Bladsjön outside of Åsbro and its existing 400 kV power line. Residential homes are marked with green outlining. The yellow lines represents the existing power lines, (NW-SE, 400 kV, 450 A), (N-S, 130 kV, -200 A). Alternative 3B is light blue, 3C is dark blue, 3D is green and 4A is purple.

The input data for this project were taken from a CAD drawing of the site, which determined the global coordinates for the towers as well as the altitude of the conduc-

tors. Also the lowest point of the lines between towers was needed in the hyperbolic approximation.

1.4.3 Grundfors

Grundfors is another site that has been calculated by WSP in a previous project and will serve as a second comparison to the calculation method. An overview of the proposed changes is shown in figure 1.4. The site is a high voltage switchgear station that is going to be moved to the location of the red box. The distance from the system to the point of interest for these calculations, which is the east house, is very large. This puts in question the relevance of the result as it clearly will be well below the limit. Still the example serves a purpose in the context of this report in comparing the threshold location relative to the power lines.

Figure 1.4: Overhead image of the site at Grundfors. The two green circles indicate houses of interest, magenta lines represent the altered section of the power lines and the red outline the new location of the switch yard. The east green line is the cross section compared using the reference material and the calculation method.

The high power switchgear station will also produce a substantial magnetic field in the area but this is a complex system to analyze and its contribution at the points is not considered. The contribution of the switchgear station is also excluded in the reference data.

amount of ferromagnetic material present [13]. There are some special conditions that need to be addressed in these calculations, however. If the ground composition has a large quantity of conductive material present, there might be a large difference between the result and the actual field strength. Results in [13] show that these effects will be very small in non-extreme cases. The results in the report only show a relative difference of under 0.1 % with a change of resistivity from 1Ωm up to 10⁵Ωm, corresponding to a span from sea water to concrete.

If there is an electrically conducting shield plate present, the magnetic field will be perturbed in the vicinity. However, this case will be disregarded in this thesis. The results here can still serve as representative of a worst case scenario.

Cables are typically buried at quite shallow depths, making for shorter distances from the conductors to the exposure area. As can be observed in the results (section 3.2), the accuracy of the calculations is dependent on the segmentation degree and the field point distance from a conductor. It is thus important to have a finer segmentation of the system near these field points. If the conductors are not segmented finely enough, the error of the calculation will be large above the conductors. The result will still be a viable approximation at some distance away from the cable, so depending on the area or point of interest, longer conductor segments may still be used. It is important to make this assessment when using the script in order to get reliable results for any given situation.

2 Methodology

2.1 Biot Savart’s law

For calculations of magnetic field contributions from power lines in 3D space, Biot Savart’s law was used. Biot Savart’s law is an integral equation that can be approximated as a sum of field contributions from straight wire segments.

~B= ^µ0I 4π

Z

C

d~_l× ~R R³ ≈ ^µ0I

4π

∑

∆~_Li× ~R_i

R³_i (2.1)

In equation 2.1 µ0is the permeability in vacuum,~R is the vector from the infinitesimal current source segment to the field point and d~l is the infinitesimal segment vector.

Conductors are approximated as line currents. The arc length vector differential d~_{l is} replaced by a directed straight line segment∆~L beginning and ending at adjacent control points on the conductor. ~_Ri is the vector pointing from the mid point of the straight segment to the field point. The error created by this approximation depends on the lengths of the straight segments as well as the position of the field point. Furthermore, the approximation of the integral is the crudest one possible, except for that~_{R is de-} termined from the center-point of each segment. An alternative way would be to use Simpson’s rule for a more accurate solution. Since the results produced here are already sufficiently accurate, this will be a future addition to the algorithm.

When considering field points close to the conductor it may be necessary to describe the source as a finite volume τ with the corresponding infinitesimal volume segment dτ containing a current density~J. The equation for Biot Savart’s law is hence changed to:

~_B= ^µ0 4π

Z

τ

(~Jdτ) × ~R

R³ (2.2)

More accurate calculations would require that the current density vary inside the con- ductors, because of the skin effect.

In this thesis, calculations are made only for field points well away from the conductors.

Furthermore, the total current in each conductor is a given entity. Therefore, it is a valid approximation to treat the conductors as being infinitesimally thin and to neglect skin effect. Propagation effects are neglected due to the low AC frequency. The current is treated as a constant along a conductor at a given instant of time. The magnetic field

The finished algorithm was tested against the simple case of a straight conductor of finite length. This case could be solved both analytically and numerically. The results were then compared to estimate the error of the approximation as a function of the distance from the source and of the segment length. The algorithm was also tested against the case of a hyperbolic hanging conductor by comparing with the numerical result at fine segmentation.

There are in theory other methods that could be used to calculate the magnetic field such as Finite Element Method (FEM). Nevertheless, this type of method would demand a large number of elements and therefore require excessive processing power.

2.2 Simplifications

In the assessed real world cases there are some approximations made to make the calculations manageable. First of all, only parts of the overall system are considered.

Only small parts of the total conductors are used, as the contributions to the magnetic field from the distant parts of the conductors are very limited. Nevertheless the total lengths of the conductors need to be large enough to include the essential contributions to the total B-field. As the field strength is largely dependent on the distance from the conductors the total lengths can be relatively small.

The rms value of the current (assumed equal for all conductors) will in reality vary over a year. Here, only the recorded yearly average of the rms line current is used. This should be regarded as a representative value.

In the real cases each phase current is split between two conductors with a separa- tion of 0.45 m. This has some influence on the B-field, at least near conductors. In the thesis, split phase conductors are replaced by single (thin) line conductors which are placed between the split conductors in space.

Induced currents in the shield wires will also have some effect on the result and may be a feature included in a future version of the script.

Furthermore, the conductors are defined as thin lines with negligible cross section area, thereby excluding the need to use current densities and also excluding the skin effect and any material parameters.

2.3 Calculation method

2.3.1 Yearly current average

The currents that are used for the calculations are yearly averages. SVK calculate these averages from a market model simulation over a year with different scenarios. These values are then compared with historical measured currents, when possible [14]. The instantaneous magnitudes are disregarded although these values can be much higher.

The three phase currents are described using complex values, all having the same absolute magnitude equal to the yearly average (RMS) current. As usual, the difference in phase angle between two line currents is ±120^◦. The yearly average current is multiplied with the phase shift that is described by:

e^j(φ±phase angle) (2.3)

to mathematically describe each phase current, φ is the phase angle of the complete three-phase circuit. This angle φ is needed to account for different loads and for the presence of several three phase circuits. The phase angle will be dependent on the load impedance and will be an input variable for the calculation method [15]. For an inductive load the current will be lagging behind the line to neutral voltage of the conductor and therefore have a negative angle. The opposite is true for a primarily capacitive load.

2.3.2 Numerical calculation method

The numerical method for calculating the B-field is based on Biot-Savart’s law for line currents, as given in 2.1. The input data are organized as 3D coordinates for the con- ductors. Two auxiliary vectors are needed for the calculation: ∆~_L

i and~_{R. ∆}~_L

i is the vector from the starting point to the ending point of the straight current (or conductor) segment.~R is the vector from the middle point of the segment to the field point. These vectors are determined for all segments of all conductors of the system. This process is independent of the shape and location of any conductor in the defined 3D space.

The magnetic field is calculated using equation 2.1 for each phase conductor using the complex currents, calculated from the yearly averages provided by WSP.

All the contributions from the segments of all phases and circuits are then superposed, i.e. added as complex-valued vectors. Thus, each spatial component(B_x, By, Bz)_{of the} magnetic field, at a given field point, has a complex value with a magnitude and an argument (phase angle). The total magnitude of the magnetic field from all conductors present is then given by the equation below:

B_total= q

(Re(B_x))²+ (Re(B_y))²+ (Re(B_z))²+ (Im(B_x))²+ (Im(B_y))²+ (Im(B_z))² (2.4)

3 Results

3.1 Special case evaluation : Straight conductor

To evaluate the calculation algorithm a base case was set up. The base case needed to be solvable both analytically and numerically. For this reason, a single thin straight, finitely long conductor was chosen. The error will be dependent on the step size or the segmentation degree of the conductor, as the vector_R~_i only will be exact for the middle point of the segment and produce a larger error at the segment ends. The longer the finite segment is, the worse the approximation of~_R

iwill be. A conductor is placed at the origin and with direction along the y-axis. The analytical calculation of the given case is presented below for a specific point in 3D space.

~_B= ^µoI 4π

Z ^L

2

−^L₂

d~_l× ~R

R³ (3.1)

Since the conductor is only stretching out in the y direction, d~_l=dy⁰ˆy, the field point is chosen as P= (50, 0, 0)or 50 m radially out from the center point of the conductor.

This choice of field point makes it possible to get an explicit analytic expression for the integral. The vector~R is determined by the differences of (unprimed) field and (primed) source coordinates.

~_R= (x−x⁰)ˆx+ (y−y⁰)ˆy+ (z−z⁰)ˆz (3.2) The calculation of the cross product d~_l× ~R for the span of the integration is:

d~_l× ~R=       

ˆx ˆy ˆz

0 dy⁰ 0

(x−x⁰) (y−y⁰) (z−z⁰)       

= (z−z⁰)dy⁰ˆx− (x−x⁰)dy⁰ˆz (3.3)

Since z, z’ and x’ all are zero during the integration, the expression d~_l× ~R can be reduced to

d~_l× ~R= −xdy⁰ˆz (3.4)

Similarly, the absolute value of~R is reduced to R=

q

(x−x⁰)²+ (y−y⁰)²+ (z−z⁰)²= q

x²+ (−y⁰)² (3.5)

It is now possible to solve the integral analytically:

µoI 4π

Z ^L

2

−^L₂

−xdy⁰ˆz

(x²+ (−y⁰)²)³² = ^µoIˆz 4πx

"

(−y⁰) px²+ (−y⁰)²

#^L₂

−^L₂

(3.6)

For the field point (x=50, y=0, z=0), and using a DC current of 400 A and a conductor length of L=500 m, the exact answer can be calculated as

4π·10⁻⁷·400·ˆz 4π·50

"

−250

p50²+ (−₂₅₀)² − √ ²⁵⁰ 50²+250²

#

= −1.5689 µT·ˆz (3.7)

This result serves as the reference as the same case is set up and solved with the numeri- cal computation algorithm. The results for some different segmentation degrees are then compared with the exact solution in table 3.1.

Table 3.1: Comparison of the exact integral and the numeric solution method for different segment lengths at field point (50,0,0).

Numerical result µT Number of segments ∆Liin m Relative error

-1.5684 250 2 0.0003

-1.5679 125 4 0.0006

-1.5662 50 10 0.0017

-1.5624 25 20 0.0041

-0.5218 5 100 0.6674

The relative error is in this case small for most segment lengths but increases significantly when∆Libecomes larger than 20 m. This comparison works best for underground ca- bles as they are more accurately approximated as straight conductors. For overhead power lines the hyperbolic conductor path between towers will make the error increase somewhat for larger segment sizes in comparison to the straight conductor. This is due to that the approximation of~_R

i will be less accurate towards the midpoint of the segments.

The procedure was also used for a field point closer to the conductor in order to deter- mine if the distance from the finite conductor to the field point will have any significant effect on the error.

-3.9871 250 2 0.0000

-3.9868 125 4 0.0001

-3.9860 50 10 0.0003

-4.0344 25 20 0.0118

-10.0853 5 100 1.5294

For this distance a striking difference can be observed. The result is more precise for larger segments but the numerical result for longer segments is larger than the reference value. The opposite is true for results in table 3.1. For a clearer representation of how the error changes depending on the number of segments, a plot of the error for the straight conductor is shown in figure 3.1:

0 5 10 15 20 25 30 35 40 45 50

Number of segments 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Relative error

Relative error for a 500 m long straight conductor using different segmentations

Figure 3.1: Plot showing how the error of the numerical result depends on wire

segmentation and how odd and even numbers of segments affect the relative error.

The values converge quickly at a very low error. An interesting aspect of the plot in figure 3.1 is that the error depends on if the segmentation is odd or even. To further explore how the odd and even segment numbers affect the results, one more test was performed, where points from 0.3 m to 100 m from the source were plotted for 25, 26, 100 and 101 segments. These plots are shown in figures 3.2, 3.3, 3.6 and 3.7.

0 10 20 30 40 50 60 70 80 90 100 Distance from source [m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Percent of excact result [%]

Ratio of numeric result and exact solution at different distances from source, 24 segments

Figure 3.2: 24 segments: the ratio between calculated numerical result and exact solution vs distance from source. The total length of the conductor is 500 m.

0 10 20 30 40 50 60 70 80 90 100

Distance from source [m]

0 5 10 15 20 25 30 35

Percent of excact result [%]

Ratio of numeric result and exact solution at different distances from source, 25 segments

Figure 3.3: 25 segments: the ratio between calculated numerical result and exact solution vs distance from source. The total length of the conductor is 500 m.

The difference is that for the even numbers of segments the approximate values are smaller than the exact values close to the source, while the opposite is true for the odd numbers of segments. Both plots also show signs of deviating from the exact solution further away from the source. The most exact result occurs around 30 m from the source.

As the point considered is located at the central orthogonal axis out from the conductor, even numbered segmentations will have a segment node in that location and odd will not. Moving closer to the conductor the∆Li relative size gets larger. For the case of odd segments this results in a larger contribution of the center∆Li.~R will in this case be smaller than the real value for the middle segment. This in turn results in a larger contribution of the middle component and a magnetic field amplitude larger than the exact result. The corresponding plots for 100 and 101 segments are shown in figure 3.6 and 3.7. The same behavior appears, but closer to the conductor, and the solution is most exact at a distance around 8 m.

0 10 20 30 40 50 60 70 80 90 100 Distance from source [m]

0 0.1 0.2 0.3 0.4 0.5 0.6

Percent of excact r

Figure 3.4: 100 segments: the ratio between calculated numerical result and exact solution vs distance from source. The total length of the conductor is 500 m.

0 10 20 30 40 50 60 70 80 90 100

Distance from source [m]

0 1 2 3 4 5

Percent of excact r

Figure 3.5: 101 segments: the ratio between calculated numerical result and exact solution vs distance from source. The total length of the conductor is 500 m.

The result suggests that depending on the distance of the field point of interest with respect to the conductor, the segment length should be chosen small enough. One im- portant aspect is that when calculating magnetic fields for underground cables, shorter segments than the standard of 5 m will be necessary in order to produce a sufficiently accurate result, because the conductor distance from the cables to the exposure zone is shorter. These cables are also more frequently used in urban environments close to the general public.

3.2 Special case evaluation: Hyperbolic conductor

For a more accurate assessment of the overhead transmission lines, a similar comparison was conducted. The exact result here cannot be easily determined through analytic integration. Instead, a simulation with very short segment lengths served as the reference.

The evaluation case is a hyperbolic line with length 500 m and its lowest point at approximately 27 m height above the ground plane. The magnitude of the B-field was calculated for field points placed at different horizontal distances to the lowest point of the hanging conductor. The reference here used 0.1 m segment length and was compared with the results for 5 m and 20 m segment length. The results are similar to those in the previous, straight conductor, examples:

0 10 20 30 40 50 60 70 80 90 100 Horizontal distance to lowest point of conductor [m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(5 m segments)/(0.1 m segments)

Ratio between 5 m and 0.1 m segment length, 100 segments

Figure 3.6: The ratio between 5 m segment length and the reference 0.1 m

0 10 20 30 40 50 60 70 80 90 100

Horizontal distance to lowest point of conductor [m]

0 5 10 15 20 25 30

(20 m segments)/(0.1 m segments)

Ratio between 20 m and 0.1 m segment length, 25 segments

Figure 3.7: The ratio between 20 m segment length and the reference 0.1 m As before, the error gets larger when the distance from the field point to the conductor is less than the length of a segment. Furthermore, the accuracy also declines slightly further away from the conductor.

Table 3.3: Comparison between the accuracy at the closest, optimal and 100 m horizontal distance from the conductor

Segment length m Distance from conductor m Ratio numeric/reference

1 0.3 0.4945

1 5 0.9999

1 100 0.9995

10 0.3 0.00749

10 20 0.9832

10 100 0.9776

Results shown in table 3.3 indicate that the hyperbolic curve influences the result in a negative way, even though the segmentation length is shorter than in the straight conductor case. When comparing the results for 20 m segments shown in figure 3.2 the peak ratio value of the solution is 0.9978 at the distance of 33 m. The peak ratio is still located further away from the conductor but the value is closer to one. This is also true for 100 m from the conductor. At the closest point of 0.3 m the hyperbolic case is more precise.

For the purpose of visualizing the field around the conductors, the central cross section of the three-phase circuit was calculated and plotted. See figure 3.8 below.

-25 -20 -15 -10 -5 0 5 10 15 20 25 x [m]

0 5 10 15 20 25 30 35 40

y[m]

Figure 3.8: Vector field plot at time t=0 for the three-phase circuit for two dimensions.

The distance between adjacent conductors is 10 m. The instantaneous currents are, from left to right, 400 A (phase 1), -200 A (phase 2), -200 A (phase 3).

Since the vector arrows are hard to see in this plot an upscaled version in 3D for three different cross sections is included in the appendix. The currents used for this plot are values for a chosen instant of time, corresponding to phase1=400 A, phase2= −200 A and phase3= −_{200 A.}

3.3 Bladsjön

The power transmission line shown in figure 1.3 (above) was analyzed. The line con- figuration analyzed is the green line named 3D. Coordinate data for the towers and lowest point of the line between towers were provided by WSP. Using these coordinates a hyperbolic function was approximated between each tower giving the 3D conductor geometry, which was split into 5 m segments as prescribed by SVK [14]. In WSP:s calculations the induction in the neutral top conductors is also taken into account. These conductors get induced currents flowing through them, currents that in turn create their own magnetic fields. These contributions to the total magnetic field are not taken into account in this report but may be added in a future version of the algorithm. The power line is a 400 kV line that carries an annual average current of 450 A, southbound power- flux. In the calculation the smaller 130 kV line labeled Bef in figure 1.3 is modeled with

an annual average current of 200 A, northbound power-flux. This line has significantly less effect on the total magnetic field at each point considered, as both the distance from the point is longer, phase conductors are located closer together and the currents through the conductors are smaller.

The calculations are done for the four points that represent residential houses in the vicinity of the power transmission system. The complete site is shown in figure 3.9.

Field contributions from the continuation of the lines in both directions (outside the figure) are very small and are not included in the reference material, as previously stated.

The (reference) phase angle for the circuits has been defined as zero in both cases.

This was also done in the reference material.

1.504 1.506 1.508 1.51 1.512 1.514

Global coordinates [m] #10⁵

6.542 6.5422 6.5424 6.5426 6.5428 6.543 6.5432

Global coordinates [m]

#10⁶ Input data plotted for site Bladsjön with house coordinates

400kV Phase1 400kV Phase2 400kV Phase3 130kV Phase1 130kV Phase2 130kV Phase3 Used Cross section House 1:74 House 1:76 House 1:75 House 1:71

Figure 3.9: 2D representation of the site Bladsjön with relevant houses. The red line corresponds to the cross section in figures 3.10 and 3.11. The black rectangle represents the calculation area used in figure 3.12

The circles represent the nearest points of the houses 1.5 m above ground. The calculated magnetic field strengths at these coordinates for cases with line segment lengths 5 m and 1 m are shown in Table 3.4

1:74 0.2553 0.2425

1:76 0.1771 0.1528

1:75 0.2760 0.2557

1:71 0.2873 0.2806

WSP:s results for house 1:71 were estimated from figure 3.10. The same cross section was plotted using the produced algorithm; this graph is shown in figure 3.11. The lower field magnitude close to the center-phase may be due to differences in line data. As can be seen in figure 3.9, the conductors are spaced uniformly over each span. The fact that the conductors in figure 3.10 in reality consist of two wires per phase separated by 0.45 m, as well as the lack of induced current in ground wires, may also affect the calculations.

Figure 3.10: Graph of the magnetic field near house 1:71 in WSP’s report for Bladsjön.

The section is indicated in Figure 3.8. The field calculation height is 1.5 m above ground level at 1:71. The origin indicates the center phase.

Figure 3.11: Calculation of the B-field for the same section as in Figure 3.10, using the algorithm developed by the author.

House 1:71 is located 84.9 m from the 400 kV center phase line and can therefore be estimated to 0.3 µT in WSP:s results, figure 3.10. However, the resolution of the graph in figure 3.10 makes it impossible to acquire a more precise reading of the field. At the same distance from the center conductor the calculated result shows a magnitude of 0.2871 µT.

The results are difficult to compare as the resolution of the calculated plot is much higher.

A mesh calculation of the magnetic field strength for a limited area around the five houses is shown in Figure 3.12. The calculation was made using a constant elevation, equal to that of house 1.71.

Figure 3.12: Graph of result for site Bladsjön for house 1:71 altitude above sea level The resolution of the mesh is 1 m and the segmentation length of the conductors is 5 m, which makes it possible to generate an accurate contour plot for the same area as

cost of precision. This needs to be taken into account when simulating a larger area.

Figure 3.13: Contour graph of three B-field magnitudes for site Bladsjön at house 1:71 height above sea level. The house 1:71 is indicated by the blue circle.

Contour levels are given in µT.

Since the calculation plane is at the elevation of house 1:71, the other houses still need to be calculated for, one by one. Since the coordinates can be exported, the simulation can be run for several elevations, and the resulting data for each elevation be extracted.

These coordinates can at a later stage be used to construct a 3D level surface for the limit B field value, a surface that in turn can be merged with CAD drawings of the site.

Another possibility is to include elevation (i.e. terrain) data for the site. The problem so far is that these coordinates need to be aligned with the mesh made in MATLAB or use a mesh taken directly from the CAD drawing of the site. These CAD drawings are very dense in data and need to be reduced (simplified), before running the B-field calculation script in MATLAB.

3.4 Grundfors

For this site the reference data were not completely correct. The solution method used did not accept z-coordinates for the towers; therefore the setup is a bit strange at a first glance. Also the provided line data are not completely accurate, as can be seen in figure 3.14. The orientations of the towers are not included and the left and right conductors were only displaced in the x-direction from the center conductor coordinates.

This results in the conductors being closer together than in the actual case along the span. Due to this, only the east cross segment was analyzed, as the 220 kV AL7 S1 circuit closest to the west house is to poorly approximated.

The global z-coordinates for the towers are not used and the whole site is therefore placed on a plane with constant elevation. The lines, however, have an offset correspond- ing to the elevation of the conductors. This produces some points, where the power lines are close to the virtual ground, which is not the case at the actual site. Field points are then displaced 1.5 m from the plane of the towers.

6.206 6.208 6.21 6.212 6.214 6.216 6.218 6.22 6.222 6.224 6.226

Global coordinates [m] #10⁵

7.2072 7.2074 7.2076 7.2078 7.208 7.2082 7.2084 7.2086 7.2088

Global coordinates [m]

#10⁶ Input data plotted for site Grundfors

220kV AL7 S1 Phase1 220kV AL7 S1 Phase2 220kV AL7 S1 Phase3 400kV UL28 S1-3 Phase1 400kV UL28 S1-3 Phase2 400kV UL28 S1-3 Phase3 400kV UL1 S1-3 Phase1 400kV UL1 S1-3 Phase2 400kV UL1 S1-3 Phase3 Used cross section House East House West

Figure 3.14: 2D representation of the site Grundfors with relevant houses. The yellow line represents the cross section in figure 3.15 and 3.16. The black rectangle represents the area used in figure 3.18.

The calculations for site Grundfors were made in the same manner as for site Bladsjön, in order to facilitate comparisons. The calculated result for the same cross section is shown in figure 3.16. However, the calculated B-field values at the positions of the houses are

limit B-field of 0.4 µT. Magnetic field strengths in both cases are quite similar. From the reference material the limit was determined to be located at 50-60 m from the center phase of the UL1 S1-3 circuit (see figure 3.14). As seen in figure 3.16 this value is a bit higher using the algorithm, with the limit located at approximately 65 m. Additional causes for the deviating result may be some line coordinate differences as well as the shield wires not being included. As observed in case Bladsjön the magnitude at the center phase is smaller than the corresponding WSP value. The shape of the curve in figure 3.15 is not showing the normal smooth curve shape at the center phase as in figures 3.10, 3.11 and 3.16. This suggests that the measurement start point might be at the phase conductor closest to the eastern house. This was not possible to determine from the provided data.

The coordinates in this case do not take into account the orientation of each tower as this is not known. This makes for shorter conductor separation at each node point between the towers. Having access to the coordinates of each phase attachment point for the towers would remove this issue.

Figure 3.15: B-field magnitude as a function of the horizontal distance from the center phase of line UL1 S1-3. The curve corresponds to the cross section towards the Eastern house in figure 3.14. Reference curve from WSP.

Figure 3.16: Cross section field strength plot for the same cross section as in figure 3.10, calculated result

A contour plot was also made, even though none of the houses were close to the maximum allowed B-field magnitude.

Figure 3.17: Contour plot for the Grundfors site. The east house is too far outside the relevant area and hence not included. The western house is shown as a blue circle. Contour levels are given in µT.

Finally, a meshed plot of a section of the site, showing the B-field magnitudes on the z-axis is shown in figure 3.18. The peaks correspond to local minima in line elevations above ground.

Figure 3.18: Calculated B-field magnitude for the Grundfors site. Peaks correspond to points, where the conductors are closest to ground. The field is calculated at 1.5 m above the ground plane. Compare this plot with figure 3.14

4 Discussion and analysis

4.1 Medical effects

Some research reports show correlations between electromagnetic fields and neurode- generative disease, in particular ALS. Although no conclusive proof has been produced at this point, strong links between domestic living conditions over the limit value 0.4 µT and childhood leukemia have been observed. The cause of the rise in childhood leukemia is still unknown and may have other causal reasons than the exposure to ELF magnetic fields. Still, the limit put in place is to be considered more than a safety precaution to calm the general public, but built on strong research data. The data presented in for example (A pooled analysis of magnetic fields and childhood leukaemia) show that limitation of domestic exposure is needed, although the cause of rise in disease is still obscure [10].

4.2 Calculation method

The numerical calculation method has clear improvement potential for future devel- opment. In this first version some some details were omitted, such as the shield wire induction that will effect the overall field. Nevertheless, no significant difference was noted in the comparisons with the previous data that included this feature, except close to the center conductor. The calculated magnetic fields were consistently slightly higher than the reference data except for close to the center phase.

The precision of the calculations mostly depends on the segmentation of the conductors, which also in turn affects the problem size. As the error is larger close to the conductor some thought needs to be put on selecting a suitable segmentation for any given situa- tion. For example, underground cables will be located closer to the 1.5 m measurement elevation above ground than overhead cables. Underground cables should therefore be subdivided into shorter segments in order to obtain sufficiently accurate B field values, in particular above the cables.

Many of the initial problems involved input data. In the Grundfors case this is es- pecially clear as the separation of the power line phases are depending on the direction of the circuit. This problem will be solved by including tower orientation in the global coordinate system or providing the connection coordinates of the phases for all the tow-